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Random Networks for Communication This page intentionally left blank Random Networks for Communication When is a network (almost) connected? How much information can it carry? How can you find a particular destination within the network? And how do you approach these questions – and others – when the network is random? The analysis of communication networks requires a fascinating synthesis of random graph theory, stochastic geometry and percolation theory to provide models for both structure and information flow. This book is the first comprehensive introduction for graduate students and scientists to techniques and problems in the field of spatial random networks. The selection of material is driven by applications arising in engineering, and the treatment is both readable and mathematically rigorous. Though mainly concerned with information-flow-related questions motivated by wireless data networks, the models developed are also of interest in a broader context, ranging from engineering to social networks, biology, and physics. Massimo Franceschetti is assistant professor of electrical and computer engineering at the University of California, San Diego. His work in communication system theory sits at the interface between networks, information theory, control, and electromagnetics. Ronald Meester is professor of mathematics at the Vrije Universiteit Amsterdam. He has published broadly in percolation theory, spatial random processes, self-organised crit- icality, ergodic theory, and forensic statistics and is the author of Continuum Percolation (with Rahul Roy) and A Natural Introduction to Probability Theory. CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial Board R. Gill (Mathematisch Instituut, Leiden University) B. D. Ripley (Department of Statistics, University of Oxford) S. Ross (Department of Industrial & Systems Engineering, University of Southern California) B. W. Silverman (St. Peter’s College, Oxford) M. Stein (Department of Statistics, University of Chicago) This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochastic applicable mathematics. The topics range from pure and applied statistics to probability theory, operations research, optimization, and mathematical pro- gramming. The books contain clear presentations of new developments in the field and also of the state of the art in classical methods. While emphasizing rigorous treatment of theoretical methods, the books also contain applications and discussions of new techniques made possible by advances in computational practice. Already published 1. Bootstrap Methods and Their Application, by A. C. Davison and D. V. Hinkley 2. Markov Chains, by J. Norris 3. Asymptotic Statistics, by A. W. van der Vaart 4. Wavelet Methods for Time Series Analysis, by Donald B. Percival and Andrew T. Walden 5. Bayesian Methods, by Thomas Leonard and John S. J. Hsu 6. Empirical Processes in M-Estimation, by Sara van de Geer 7. Numerical Methods of Statistics, by John F. Monahan 8. A User’s Guide to Measure Theoretic Probability, by David Pollard 9. The Estimation and Tracking of Frequency, by B. G. Quinn and E. J. Hannan 10. Data Analysis and Graphics using R, by John Maindonald and W. John Braun 11. Statistical Models, by A. C. Davison 12. Semiparametric Regression, by D. Ruppert, M. P. Wand, R. J. Carroll 13. Exercises in Probability, by Loic Chaumont and Marc Yor 14. Statistical Analysis of Stochastic Processes in Time, by J. K. Lindsey 15. Measure Theory and Filtering, by Lakhdar Aggoun and Robert Elliott 16. Essentials of Statistical Inference, by G. A. Young and R. L. Smith 17. Elements of Distribution Theory, by Thomas A. Severini 18. Statistical Mechanics of Disordered Systems, by Anton Bovier 19. The Coordinate-Free Approach to Linear Models, by Michael J. Wichura 20. Random Graph Dynamics, by Rick Durrett 21. Networks, by Peter Whittle 22. Saddlepoint Approximations with Applications, by Ronald W. Butler 23. Applied Asymptotics, by A. R. Brazzale, A. C. Davison and N. Reid Random Networks for Communication From Statistical Physics to Information Systems Massimo Franceschetti and Ronald Meester CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521854429 © M. Franceschetti and R. Meester 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-37855-3 eBook (NetLibrary) ISBN-13 978-0-521-85442-9 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page ix List of notation xi 1 Introduction 1 1.1 Discrete network models 3 1.1.1 The random tree 3 1.1.2 The random grid 5 1.2 Continuum network models 6 1.2.1 Poisson processes 6 1.2.2 Nearest neighbour networks 10 1.2.3 Poisson random connection networks 11 1.2.4 Boolean model networks 12 1.2.5 Interference limited networks 13 1.3 Information-theoretic networks 14 1.4 Historical notes and further reading 15 2 Phase transitions in infinite networks 17 2.1 The random tree; infinite growth 17 2.2 The random grid; discrete percolation 21 2.3 Dependencies 29 2.4 Nearest neighbours; continuum percolation 31 2.5 Random connection model 37 2.6 Boolean model 48 2.7 Interference limited networks 51 2.7.1 Mapping on a square lattice 56 2.7.2 Percolation on the square lattice 58 2.7.3 Percolation of the interference model 62 2.7.4 Bound on the percolation region 63 2.8 Historical notes and further reading 66 vi Contents 3 Connectivity of finite networks 69 3.1 Preliminaries: modes of convergence and Poisson approximation 69 3.2 The random grid 71 3.2.1 Almost connectivity 71 3.2.2 Full connectivity 72 3.3 Boolean model 77 3.3.1 Almost connectivity 78 3.3.2 Full connectivity 81 3.4 Nearest neighbours; full connectivity 88 3.5 Critical node lifetimes 92 3.6 A central limit theorem 98 3.7 Historical notes and further reading 98 4 More on phase transitions 100 4.1 Preliminaries: Harris–FKG Inequality 100 4.2 Uniqueness of the infinite cluster 101 4.3 Cluster size distribution and crossing paths 107 4.4 Threshold behaviour of fixed size networks 114 4.5 Historical notes and further reading 119 5 Information flow in random networks 121 5.1 Information-theoretic preliminaries 121 5.1.1 Channel capacity 122 5.1.2 Additive Gaussian channel 124 5.1.3 Communication with continuous time signals 127 5.1.4 Information-theoretic random networks 129 5.2 Scaling limits; single source–destination pair 131 5.3 Multiple source–destination pairs; lower bound 136 5.3.1 The highway 138 5.3.2 Capacity of the highway 139 5.3.3 Routing protocol 142 5.4 Multiple source–destination pairs; information-theoretic upper bounds 146 5.4.1 Exponential attenuation case 148 5.4.2 Power law attenuation case 151 5.5 Historical notes and further reading 155 6 Navigation in random networks 157 6.1 Highway discovery 157 6.2 Discrete short-range percolation (large worlds) 159 6.3 Discrete long-range percolation (small worlds) 161 6.3.1 Chemical distance, diameter, and navigation length 162 6.3.2 More on navigation length 167 6.4 Continuum long-range percolation (small worlds) 171 Contents vii 6.5 The role of scale invariance in networks 181 6.6 Historical notes and further reading 182 Appendix 185 A.1 Landau’s order notation 185 A.2 Stirling’s formula 185 A.3 Ergodicity and the ergodic theorem 185 A.4 Deviations from the mean 187 A.5 The Cauchy–Schwartz inequality 188 A.6 The singular value decomposition 189 References 190 Index 194 Preface What is this book about, and who is it written for? To start with the first question, this book introduces a subject placed at the interface between mathematics, physics, and information theory of systems. In doing so, it is not intended to be a comprehensive monograph and collect all the mathematical results available in the literature, but rather pursues the more ambitious goal of laying the foundations. We have tried to give emphasis to the relevant mathematical techniques that are the essential ingredients for anybody interested in the field of random networks. Dynamic coupling, renormalisation, ergodicity and deviations from the mean, correlation inequalities, Poisson approximation, as well as some other tricks and constructions that often arise in the proofs are not only applied, but also discussed with the objective of clarifying the philosophy behind their arguments. We have also tried to make available to a larger community the main mathematical results on random networks, and to place them into a new communication theory framework, trying not to sacrifice mathematical rigour. As a result, the choice of the topics was influenced by personal taste, by the willingness to keep the flow consistent, and by the desire to present a modern, communication-theoretic view of a topic that originated some fifty years ago and that has had an incredible impact in mathematics and statistical physics since then. Sometimes this has come at the price of sacrificing the presentation of results that either did not fit well in what we thought was the ideal flow of the book, or that could be obtained using the same basic ideas, but at the expense of highly technical complications. One important topic that the reader will find missing, for example, is a complete treatment of the classic Erdös–Rényi model of random graphs and of its more recent extensions, including preferential attachment models used to describe properties of the Internet.
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